derivative (7x)/(8-tan(x))

解答

导数

\frac{7 x}{8 - tan ( x )}

\frac{7 ( 8 - tan ( x ) + x sec ^{2} ( x ) )}{( 8 - tan ( x ) ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{7 x}{8 - tan ( x )})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 7 \frac{d}{d x} (\frac{x}{8 - tan ( x )})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= 7 \cdot \frac{\frac{d x}{d x} ( 8 - tan ( x ) ) - \frac{d}{d x} ( 8 - tan ( x ) ) x}{( 8 - tan ( x ) ) ^{2}}

\frac{d x}{d x} = 1

\frac{d}{d x} (8 - tan (x)) = - sec^{2} (x)

= 7 \cdot \frac{1 \cdot ( 8 - tan ( x ) ) - ( - sec ^{2} ( x ) ) x}{( 8 - tan ( x ) ) ^{2}}

化简

7 \cdot \frac{1 \cdot ( 8 - tan ( x ) ) - ( - sec ^{2} ( x ) ) x}{( 8 - tan ( x ) ) ^{2}} : \frac{7 ( 8 - tan ( x ) + x sec ^{2} ( x ) )}{( 8 - tan ( x ) ) ^{2}}

= \frac{7 ( 8 - tan ( x ) + x sec ^{2} ( x ) )}{( 8 - tan ( x ) ) ^{2}}

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